Then \(f_{\mathcal {L},\zeta }(\alpha )=\tau _{\mathcal {L},\zeta }^*(\alpha )\)  for each \(\alpha \in \) .

We first observe the following standard counting lemma, which is similar to [1]}, but the proof is easier.

Suppose \(\mathcal {L}\)  is any loop class (not necessarily irreducible) and the derivative \(\tau _{\mathcal {L},\zeta }^{\prime }(q)=\alpha \)  exists.
Then for any \(\delta >0\) , there is \(t_0=t_0(\delta ,q)\)  such that for all \(0<t<t_0\) , there is \(F^*(t)\subset \mathcal {F}_{\mathcal {L},\zeta }(t)\)  such that
